Splitting Blender animation into separate clips
How to create a new action


Press the F-Button so it is darkened to keep your existing action in the blend file even when you save and reload it. Otherwise you might loose your animation. So in case you're uncertain make a backup copy.
Press the  button to unlink the action. 
Press the New button to create a new action.


Copy frames from one action to another


Switch between your actions by pressing the  button and choose your desired action from the dropdown menu
Select a frame range by pressing the B key to activate the border select and select a range of frames in the dopesheet editor.
Press Ctrl + C or press the  button in the header menu to copy frames to a clipboard
Press Ctrl + D or press the  button in the header menu to paste the frames at the current cursor location (green line)